Question

1. list two triangles whose lengths can be represented by the equivalent ratios \\(\frac{4}{8} = \frac{5}{10} = \frac{3}{6}\\).

Explanation:

Step1: Simplify the ratios

First, simplify each ratio. For \(\frac{4}{8}\), divide numerator and denominator by 4: \(\frac{4\div4}{8\div4}=\frac{1}{2}\). For \(\frac{5}{10}\), divide numerator and denominator by 5: \(\frac{5\div5}{10\div5}=\frac{1}{2}\). For \(\frac{3}{6}\), divide numerator and denominator by 3: \(\frac{3\div3}{6\div3}=\frac{1}{2}\). So all ratios simplify to \(\frac{1}{2}\), meaning the triangles are similar with a scale factor.

Step2: Define the first triangle

Let's take the first triangle with side lengths 4, 5, 3. Wait, actually, the ratios correspond to sides of triangles. Let's consider the sides as per the ratios. Let the first triangle have sides \(a = 4\), \(b = 5\), \(c = 3\).

Step3: Define the second triangle

Since the ratio is \(\frac{1}{2}\) (wait, no, actually the simplified ratio is \(\frac{1}{2}\), so to get a similar triangle, we can multiply each side by 2. So the second triangle would have sides \(4\times2 = 8\), \(5\times2 = 10\), \(3\times2 = 6\). Wait, but actually, the given ratios are \(\frac{4}{8}=\frac{5}{10}=\frac{3}{6}\), so the two triangles can be: Triangle 1 with sides 4, 5, 3 and Triangle 2 with sides 8, 10, 6. Let's check the ratios: \(\frac{4}{8}=\frac{1}{2}\), \(\frac{5}{10}=\frac{1}{2}\), \(\frac{3}{6}=\frac{1}{2}\), so they are similar (by SSS similarity criterion since corresponding sides are in proportion). Alternatively, we can have other triangles. For example, Triangle A: sides 4, 5, 3; Triangle B: sides 8, 10, 6. Or we can scale by another factor, but the simplest is using the given ratios. So two triangles can be a triangle with sides 4, 5, 3 and a triangle with sides 8, 10, 6.

Answer:

One triangle has side lengths \(4\), \(5\), \(3\) and another triangle has side lengths \(8\), \(10\), \(6\) (other valid similar triangles can also be formed by scaling these sides by the same non - zero factor).